论文标题
椭圆形的R-Matrices和Feigin和Odesskii的椭圆形代数
Elliptic R-matrices and Feigin and Odesskii's elliptic algebras
论文作者
论文摘要
Feigin和Odesskii引入的代数$ q_ {n，k}（e，τ）$作为四维sklyanin代数的概括，形成了由二次代数的家族，由二次代数由cocrime Integers $ n> k \ ge 1 $ $ n> k \ ge 1 $，一个复杂的eLlipt elliptic curve $ e $ $ e $ $ $ $，以及一个点。本文的主要结果是，$ q_ {n，k}（e，τ）$具有与$ n $变量上的多项式环相同的希尔伯特系列，当$ n $变量不是$τ$不是扭转点时。我们还表明，$ q_ {n，k}（e，τ）$是一个koszul代数，因此，当$τ$不是扭转点时，全球尺寸$ n $的原因，对于所有$τ$而言，它是Artin-Schelter的常规。证明使用以下事实：定义$ q_ {n，k}（e，τ）$的二次关系空间是属于操作员家族的操作员$r_τ（τ）$的图像 $R_τ(z):\mathbb{C}^n\otimes\mathbb{C}^n\to\mathbb{C}^n\otimes\mathbb{C}^n$, $z\in\mathbb{C}$, that (we will show) satisfy the quantum Yang-Baxter equation with spectral parameter.
The algebras $Q_{n,k}(E,τ)$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers $n>k\ge 1$, a complex elliptic curve $E$, and a point $τ\in E$. The main result in this paper is that $Q_{n,k}(E,τ)$ has the same Hilbert series as the polynomial ring on $n$ variables when $τ$ is not a torsion point. We also show that $Q_{n,k}(E,τ)$ is a Koszul algebra, hence of global dimension $n$ when $τ$ is not a torsion point, and, for all but countably many $τ$, it is Artin-Schelter regular. The proofs use the fact that the space of quadratic relations defining $Q_{n,k}(E,τ)$ is the image of an operator $R_τ(τ)$ that belongs to a family of operators $R_τ(z):\mathbb{C}^n\otimes\mathbb{C}^n\to\mathbb{C}^n\otimes\mathbb{C}^n$, $z\in\mathbb{C}$, that (we will show) satisfy the quantum Yang-Baxter equation with spectral parameter.
